Two-loop calculation of anomalous kinetics of the reaction $A + A \rightarrow\varnothing$ in randomly stirred fluid

TL;DR

This paper investigates how random fluid flows influence the long-term behavior of the annihilation reaction A + A -> 0, using advanced two-loop renormalization group techniques to account for velocity and density fluctuations.

Contribution

It provides a two-loop renormalization group analysis of the reaction in a stochastic fluid, revealing the impact of velocity fluctuations on the reaction rate at next-to-leading order.

Findings

Velocity fluctuations affect the reaction constant at two-loop order.

Stable fixed points are identified in the two-parameter expansion.

A renormalized integro-differential equation for density is derived.

Abstract

The single-species annihilation reaction $A + A \rightarrow\varnothing$ is studied in the presence of a random velocity field generated by the stochastic Navier-Stokes equation. The renormalization group is used to analyze the combined influence of the density and velocity fluctuations on the long-time behavior of the system. The direct effect of velocity fluctuations on the reaction constant appears only from the two- loop order, therefore all stable fixed points of the renormalization group and their regions of stability are calculated in the two-loop approximation in the two-parameter $(\epsilon, \Delta)$ expansion. A renormalized integro- differential equation for the number density is put forward which takes into account the effect of density and velocity fluctuations at next-to-leading order. Solution of this equation in perturbation theory is calculated in a homogeneous system.